3. THE DENSITY PARAMETER

3.1. Model-Independent Determinations

3.1.1. Luminosity Density x < M/L >

The mass density can be obtained by multiplying the luminosity density
with galaxy's average mass to light ratio < M/L >.
The local luminosity density, evaluated by integrating the luminosity function,
is reasonably well converged to
LB = (2.0 ± 0.4) x 108hL
Mpc-3 from many observations.
The M/LB of galaxies generally increases with the scale.
When the mass is integrated to
100 kpc, a typical
M/LB is about (100-200)h
in solar units, and it may still increase outward (e.g.,
Faber & Gallagher
1979;
Little & Tremaine
1987;
Kochanek 1996;
Bahcall et al. 1995;
Zaritsky et
al. 1997).
The virial radius in a spherical collapse model is
r = 0.13 Mpc -0.15
[M / 1012M]1/2< 100 kpc.
If the dark matter distribution is isothermal within the virial radius,
the value of M/LB inside the virial radius is
(150-400)h for L* galaxies.
This is about the value of M/LB for groups and clusters,
(150-500)h.
Multiplying the two values we get
= 0.20 x 2±1.
See also
Fukugita, Hogan &
Peebles (1998)
for variants of this argument.

The important assumption for these calculations is the absence
of copious matter outside the clusters. This is a question difficult
to answer, but the observation of weak lensing
around the clusters indicate that the distributions of dark mass
and galaxies are similar at least in the vicinity of clusters
(Tyson & Fischer
1995;
Squires et
al. 1996).

Some attempts have also been made to estimate the mass on a supercluster scale.
Small et al. (1998)
inferred M / LB 560h for the Corona Borearis
supercluster, by applying
the virial theorem (inspired by an N body simulation). On the other
hand,
Kaiser et al. (1998)
estimated M / LB 250 from a mesurement
of the gravitational shear of weak lensing caused by a supercluster
MS0302+17(3);
the result is not well convergent, but it seems unlikely that
is larger than 0.5.

3 They suggest
0.04 on the basis that only
early-type galaxy population traces the mass distribution and the luminosity
density is multiplied by the fraction of early-type galaxies (20%).
It seems possible that late type galaxies reside in low density
regions, causing only a small shear, which is buried in noise,
and escaped from the measurement. Back.